For a face-edge path \(\pi \) , we let \(\Gamma (\pi )=\emptyset \)  if \(\pi \)  has exactly one edge of \(G_\textsf {mod}\) , and it is is constructed due to the radial completion of \(G\) ,
and let \(\Gamma (\pi )=\Sigma \cup \Sigma ^{-1} \cup \lbrace \epsilon \rbrace \) , otherwise. Recall that the edges of \(G_\textsf {mod}\)  constructed due to the radial
compleition of \(G\)  lie on the frames only, and thus the number of such edges is \(2^{O(k)}\) .
This is the exactly same was as the definition of \(\Gamma \)  in [1]}.
Then Schrijver showed that a flow function \(\psi \)  with \(\psi (\pi )\in \Gamma (\pi )\)  corresponds to a linkage.
Here, we additionally set \(\Gamma (\pi )\)  for a face-edge path \(\pi \)  in \(\Pi _\textsf {frame}\) .
Recall that the crossing pattern tells us the sequence of the crossings between each frame and the paths of our target linkage \(\mathcal {P}\) 
sorted along the frame. In fact, we construct the weak linkage \(\mathcal {W}\)  in Section 
so that the walks of \(\mathcal {W}\)  and \(\mathcal {W}_\textsf {mod}\)  cross each frame in the same order as the paths of \(\mathcal {P}\) .
See Figure REF (c).
For each face-path \(\pi \)  in \(\Pi _\textsf {frame}\) , we let \(\Gamma (\pi )=\lbrace \phi (\pi )\rbrace \) . Recall that the flow function \(\phi \)  is corresponding the canonical weak linkage \(\mathcal {W}_\textsf {mod}\)  in \(G_\textsf {mod}\) .
